📝 Chapter Notes & Revision

Comparing Quantities

🏫 MP BoardClass 8Mathematics

📐 Formula & Cheat Sheet (English)

Quick Revision Notes & Formula Sheet

Class 8 Mathematics — Chapter: Comparing Quantities (राशियों की तुलना)

1. Ratios and Percentages (अनुपात और प्रतिशत)

  • Ratio (अनुपात): A comparison of two quantities of the same kind measured in the same units.
    • Ratio of $a$ to $b$ is written as $a : b$ or $\frac{a}{b}$.
  • Percentage (प्रतिशत): A fraction with denominator 100. Means "per hundred". Symbol: %.
  • Converting Fraction / Ratio to Percentage: $$\text{Percentage} = \text{Fraction or Ratio} \times 100$$
  • Converting Percentage to Fraction: $$\text{Fraction} = \frac{\text{Percentage}}{100}$$

2. Increase and Decrease Percentage (वृद्धि और कमी प्रतिशत)

  • Percentage Increase: $$\text{Percentage Increase} = \left( \frac{\text{Amount of Increase}}{\text{Original Value}} \right) \times 100$$

  • Percentage Decrease: $$\text{Percentage Decrease} = \left( \frac{\text{Amount of Decrease}}{\text{Original Value}} \right) \times 100$$

  • New Value: $$\text{New Value} = \text{Original Value} \pm \text{Change Amount}$$

3. Discount (बट्टा / छूट)

  • Marked Price (MP / अंकित मूल्य): The price printed on an item.

  • Selling Price (SP / विक्रय मूल्य): The price at which the item is actually sold.

  • Discount Formula: $$\text{Discount} = \text{Marked Price (MP)} - \text{Selling Price (SP)}$$

  • Discount Percentage: $$\text{Discount %} = \left( \frac{\text{Discount}}{\text{Marked Price}} \right) \times 100$$

  • Finding SP using Discount %: $$\text{SP} = \text{MP} \times \frac{(100 - \text{Discount %})}{100}$$

4. Profit and Loss (लाभ और हानि)

  • Cost Price (CP / क्रय मूल्य): The buying price of an item plus overhead expenses (transportation, repair, etc.).
  • Selling Price (SP / विक्रय मूल्य): The price at which the item is sold.

A. Profit (Profit happens when SP > CP)

  • $\text{Profit} = \text{SP} - \text{CP}$
  • $\text{Profit %} = \left( \frac{\text{Profit}}{\text{CP}} \right) \times 100$
  • $\text{SP} = \text{CP} \times \frac{(100 + \text{Profit %})}{100}$
  • $\text{CP} = \frac{\text{SP} \times 100}{100 + \text{Profit %}}$

B. Loss (Loss happens when CP > SP)

  • $\text{Loss} = \text{CP} - \text{SP}$
  • $\text{Loss %} = \left( \frac{\text{Loss}}{\text{CP}} \right) \times 100$
  • $\text{SP} = \text{CP} \times \frac{(100 - \text{Loss %})}{100}$
  • $\text{CP} = \frac{\text{SP} \times 100}{100 - \text{Loss %}}$

Note: Profit % and Loss % are always calculated on the Cost Price (CP) unless specified otherwise.

5. Taxes: Sales Tax / VAT / GST

  • Taxes are added to the selling price of the item before calculating the total bill.
  • Formula: $$\text{Tax Amount} = \left( \frac{\text{Tax %}}{100} \right) \times \text{Selling Price}$$ $$\text{Total Bill Amount} = \text{Selling Price} + \text{Tax Amount}$$

6. Simple Interest (SI) vs Compound Interest (CI)

Where:

  • $P$ = Principal (मूलधन)
  • $R$ = Rate of interest per annum (ब्याज दर % वार्षिक)
  • $T$ or $n$ = Time period in years (समय)
  • $A$ = Total Amount (मिश्रधन)

A. Simple Interest (साधारण ब्याज)

Interest calculated only on the principal amount throughout the period.

  • $\text{SI} = \frac{P \times R \times T}{100}$
  • $\text{Amount (A)} = P + \text{SI}$

B. Compound Interest (चक्रवृद्धि ब्याज)

Interest calculated on the accumulated principal plus interest of previous periods.

  • 1. Compounded Annually (वार्षिक संयोजन): $$A = P \left(1 + \frac{R}{100}\right)^n$$ $$\text{CI} = A - P$$

  • 2. Compounded Half-Yearly (अर्द्धवार्षिक संयोजन):

    • Time becomes double ($2n$)
    • Rate becomes half ($\frac{R}{2}$) $$A = P \left(1 + \frac{R}{200}\right)^{2n}$$

  • 3. Compounded Quarterly (त्रैमासिक संयोजन):

    • Time becomes four times ($4n$)
    • Rate becomes one-fourth ($\frac{R}{4}$) $$A = P \left(1 + \frac{R}{400}\right)^{4n}$$

7. Applications of Compound Interest Formula

A. Growth (Population / Bacteria Growth)

If population grows at a rate of $R%$ per year: $$\text{Population after } n \text{ years} = P_0 \left(1 + \frac{R}{100}\right)^n$$ (Where $P_0$ is initial population)

B. Depreciation (अवमूल्यन / घिसावट)

Decrease in the value of an asset (like machinery, vehicles) over time at rate $R%$: $$\text{Value after } n \text{ years} = V_0 \left(1 - \frac{R}{100}\right)^n$$ (Where $V_0$ is initial value)

Quick Memory Tricks for MP Board Exams:

  1. Overhead Charges: Always add repair, transport, or labor costs to CP to get the Total CP.
  2. Discount is on MP, Profit/Loss is on CP, and Tax is on SP.
  3. In Half-Yearly problems: Double the time, Half the rate.